Question

In matrix equation [A]{X} = {R},

\(\left[ {\rm{A}} \right] = \left[ {\begin{array}{*{20}{c}} 4&8&4\\ 8&{16}&{ - 4}\\ 4&{ - 4}&{15} \end{array}} \right],\;\left\{ X \right\} = \left\{ {\begin{array}{*{20}{c}} 2\\ 1\\ 4 \end{array}} \right\}\;and\;\left\{ R \right\} = \left\{ {\begin{array}{*{20}{c}} {32}\\ {16}\\ {64} \end{array}} \right\}.\)

One of the eigenvalues of matrix [A] is

1. 4
2. 8
3. 15
4. 16

Answer 1

Correct Answer - Option 4 : 16

Concept:

• The roots of characteristic equation |A - λI| = 0 are known as Eigen values of matrix A.
• To each Eigen value of λ if there exists a non-zero vector X such that AX = λX then X is called Eigen vector of matrix A corresponding to the Eigen value λ.

Calculation:

Given:

 \(\left[ A \right] = \left[ {\begin{array}{*{20}{c}} 4&8&4\\ 8&{16}&{ - 4}\\ 4&{ - 4}&{15} \end{array}} \right],\;\left\{ x \right\} = \left\{ {\begin{array}{*{20}{c}} 2\\ 1\\ 4 \end{array}} \right\}\;and\;\left\{ R \right\} = \left\{ {\begin{array}{*{20}{c}} {32}\\ {16}\\ {64} \end{array}} \right\}\)

\(\left[ A \right]\left\{ x \right\} = \left\{ R \right\} = \left[ {\begin{array}{*{20}{c}} 4&8&4\\ 8&{16}&{ - 4}\\ 4&{ - 4}&{15} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} 2\\ 1\\ 4 \end{array}} \right\} = \left\{ {\begin{array}{*{20}{c}} {32}\\ {16}\\ {64} \end{array}} \right\} = 16\left\{ {\begin{array}{*{20}{c}} 2\\ 1\\ 4 \end{array}} \right\}\)

This is in the form of AX = λX, where λ is Eigenvalue.

∴ One of the Eigenvalue of matrix [A] is 16